Fast matrix multiplication algorithms are asymptotically faster than the classical cubic-time algorithm, but they are often slower in practice. One important obstacle is the use of complex coefficients, which increases arithmetic overhead and limits practical efficiency. This paper focuses on transforming complex-coefficient matrix multiplication schemes into equivalent real- or rational-coefficient ones. We present a systematic method that, given a complex-coefficient scheme, either constructs a family of equivalent rational algorithms or proves that no equivalent rational scheme exists. Our approach relies only on basic linear-algebraic properties of similarity transformations of complex matrices. This method recovers the previously known ad hoc results of Dumas, Pernet, and Sedoglavic (2025) and extends them to more general settings, including algorithms involving rational coefficients and square roots, with $i=\sqrt{-1}$ as a special case. Using this framework, we show that no rational scheme is equivalent to Smirnov's $\langle4,4,9,104\rangle$ $\mathbb{Q}[\sqrt{161}]$ algorithm (2022) and that no real scheme is equivalent to the $\langle4,4,4,48\rangle$ complex algorithm of Kaporin (2024). More generally, our approach can also be used to prove the non-existence of integer-coefficient schemes.

翻译：快速矩阵乘法算法在渐近意义上比经典的三次时间算法更快，但在实践中往往更慢。一个重要的障碍是复数系数的使用，这会增加算术开销并限制实际效率。本文专注于将复数系数矩阵乘法方案转化为等价的实数或有理数系数方案。我们提出了一种系统方法，给定一个复数系数方案，该方法要么构造一族等价的有理数算法，要么证明不存在等价的复数方案。我们的方法仅依赖于复数矩阵相似变换的基本线性代数性质。此方法恢复了 Dumas、Pernet 和 Sedoglavic (2025) 先前已知的特定结果，并将其推广到更一般的设置，包括涉及有理数系数和平方根的算法，其中 $i=\sqrt{-1}$ 作为一个特例。利用此框架，我们证明了不存在等价于 Smirnov (2022) 的 $\langle4,4,9,104\rangle$ $\mathbb{Q}[\sqrt{161}]$ 算法的有理数方案，也不存在等价于 Kaporin (2024) 的 $\langle4,4,4,48\rangle$ 复数算法的实数方案。更一般地，我们的方法也可用于证明整数系数方案的不存在性。